Simulations of estimating causal effects using undersmoothed HAL
Seraphina Shi
2023-03-17
Simulation #1
Data structure: \(O = (W, A, Z, Y)\)
- U - exogenous variables
- W - baseline covariate that is a measure of body condition
- A - treatment level based on W, continuous between 0 and 5
- Z - intermediate curve based on W and A
- Y - outcome, indicator of an event ?
Underlying data generating process, \(P_{U,X}\)
- Exogenous variables:
- \(U_A \sim Normal(\mu=0, \sigma^2 =
1^2)\)
- \(U_A \sim Normal(\mu=0, \sigma^2 =
2^2)\)
- \(U_Z \sim Uniform(min = 0, max =
1)\)
- \(U_Y \sim Uniform(min = 0, max = 1)\)
- \(U_A \sim Normal(\mu=0, \sigma^2 =
1^2)\)
- Structural equations F and endogenous variables:
- \(W = U_W\)
- \(A = bound(2 - 0.5W + U_A, min=0,
max=5)\)
- \(Z = \mathbf{I}[U_Z < expit(2-W-A)]\)
- \(Y = \mathbf{I}[U_Y < expit(W + 5*A + Z - 0.5 * W * A - 8)]\)
- \(W = U_W\)
## W A Z Y
## Min. :-3.507788 Min. :0.0000 Min. :0.0000 Min. :0.0000
## 1st Qu.:-0.667617 1st Qu.:0.5639 1st Qu.:0.0000 1st Qu.:0.0000
## Median : 0.001434 Median :1.9834 Median :0.0000 Median :1.0000
## Mean :-0.000198 Mean :2.0947 Mean :0.4967 Mean :0.5958
## 3rd Qu.: 0.660323 3rd Qu.:3.3522 3rd Qu.:1.0000 3rd Qu.:1.0000
## Max. : 3.531910 Max. :5.0000 Max. :1.0000 Max. :1.0000## Summary of A given W < -1:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 1.325 2.711 2.660 4.099 5.000## Summary of A given -1 < W <= 0:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0000 0.7848 2.1259 2.2182 3.4684 5.0000## Summary of A given 0 < W <= 1:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0000 0.4076 1.8470 1.9644 3.2074 5.0000## Summary of A given 1 < W:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 0.000 1.302 1.550 2.519 5.000
n = 200
CV HAL
results
## CV selected lambda (from one sample): 0.00669857527625325
1000 repetition
## The average of CV selected lambdas (from 1000 sample): 0.00805053386604685 The average of CV selected lambdas (from 1000 sample): 0.00805577329430703 The average of CV selected lambdas (from 1000 sample): 0.00805745619998476## z=1:
## z=0:
Globally Undersmoothed HAL
results
## Undersmoothed lambda: 0.0015646006085061
## which is 1 * lambda_CV
1000 repetition
## The average of unsersmoothed lambda (from 1000 sample): 0.00198207951272976
## which is 1 * the average of 1000 lambda_CV The average of unsersmoothed lambda (from 1000 sample): 0.00198377327801413
## which is 1 * the average of 1000 lambda_CV## z=1:
## z=0:
Oevr a grid of lambda scalers
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.3844]
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.4120]
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.4396]n = 50
CV HAL
results
## CV selected lambda (from one sample): 0.0195155536662222
1000 repetition
## The average of CV selected lambdas (from 1000 sample): 0.0212969848400777 The average of CV selected lambdas (from 1000 sample): 0.0213398235684669 The average of CV selected lambdas (from 1000 sample): 0.0212988488932532 The average of CV selected lambdas (from 1000 sample): 0.0213307346613431## z=1:
## z=0:
Globally Undersmoothed HAL
results
## Undersmoothed lambda: 0.00943173094359552
## which is 1 * lambda_CV
1000 repetition
## The average of unsersmoothed lambda (from 1000 sample): 0.00552028249707928
## which is 1 * the average of 1000 lambda_CV The average of unsersmoothed lambda (from 1000 sample): 0.00549945830564472
## which is 1 * the average of 1000 lambda_CV The average of unsersmoothed lambda (from 1000 sample): 0.00551185563084549
## which is 1 * the average of 1000 lambda_CV The average of unsersmoothed lambda (from 1000 sample): 0.00550421652997886
## which is 1 * the average of 1000 lambda_CV## z=1:
## z=0:
Oevr a grid of lambda scalers
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.8162]
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.8438]
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.8714]n = 500
CV HAL
results
## CV selected lambda (from one sample): 0.00304915463126771
1000 repetition
## The average of CV selected lambdas (from 1000 sample): 0.003778051893939 The average of CV selected lambdas (from 1000 sample): 0.00377959561096255## z=1:
## z=0:
Globally Undersmoothed HAL
results
## Undersmoothed lambda: 0.000239285449803067
## which is 1 * lambda_CV
1000 repetition
## The average of unsersmoothed lambda (from 1000 sample): 0.000976521442732073
## which is 1 * the average of 1000 lambda_CV## z=1:
## z=0:
Simulation 2
## W A Z Y
## Min. :-3.845320 Min. :0.0000 Min. :0.0000 Min. :0.0000
## 1st Qu.:-0.667969 1st Qu.:0.6041 1st Qu.:0.0000 1st Qu.:0.0000
## Median :-0.011089 Median :1.9688 Median :0.0000 Median :0.0000
## Mean :-0.002372 Mean :2.0967 Mean :0.4956 Mean :0.4335
## 3rd Qu.: 0.673347 3rd Qu.:3.3664 3rd Qu.:1.0000 3rd Qu.:1.0000
## Max. : 3.847768 Max. :5.0000 Max. :1.0000 Max. :1.0000## Summary of A given W < -1:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 1.378 2.682 2.670 4.053 5.000## Summary of A given -1 < W <= 0:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0000 0.8273 2.1893 2.2600 3.5571 5.0000## Summary of A given 0 < W <= 1:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0000 0.4312 1.7657 1.9311 3.1612 5.0000## Summary of A given 1 < W:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 0.000 1.231 1.511 2.510 5.000
n = 200
CV HAL
results
## CV selected lambda (from one sample): 0.00342499231220454
1000 repetition
## The average of CV selected lambdas (from 1000 sample): 0.00388328921066168 The average of CV selected lambdas (from 1000 sample): 0.00387821237212883## z=1:
## z=0:
Globally Undersmoothed HAL
results
## Undersmoothed lambda: 0.000799982807508477
## which is 1 * lambda_CV
1000 repetition
## The average of unsersmoothed lambda (from 1000 sample): 0.00085245114733228
## which is 1 * the average of 1000 lambda_CV The average of unsersmoothed lambda (from 1000 sample): 0.000851916105948073
## which is 1 * the average of 1000 lambda_CV## z=1:
## z=0:
Oevr a grid of lambda scalers
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.16031]
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.16307]
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.16583]
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.16859]
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.17135]
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.17411]
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.17687]
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.17963]n = 50
CV HAL
results
## CV selected lambda (from one sample): 0.0198714531342128
1000 repetition
## The average of CV selected lambdas (from 1000 sample): 0.0196135234410873 The average of CV selected lambdas (from 1000 sample): 0.0195494261017948 The average of CV selected lambdas (from 1000 sample): 0.0195701963312569## z=1:
## z=0:
Globally Undersmoothed HAL
results
## Undersmoothed lambda: 0.00943173090927282
## which is 1 * lambda_CV
1000 repetition
Since the sample size n=50 could be too small for undersmoothing, simulations returned NA as the unsersmoothed lambda for many rounds. So I just skipped this part.
n = 500
CV HAL
results
## CV selected lambda (from one sample): 0.0016481890921567
1000 repetition
## The average of CV selected lambdas (from 1000 sample): 0.00184274700569968## z=1:
## z=0:
Globally Undersmoothed HAL
results
## Undersmoothed lambda: 0.000267629110962862
## which is 1 * lambda_CV
1000 repetition
## The average of unsersmoothed lambda (from 1000 sample): 0.000393879557187377
## which is 1 * the average of 1000 lambda_CV## z=1:
## z=0:
